Groupoid objects in the category of derived manifolds

Question

I am learning about "derived manifolds" from talks of Ping Xu in Higher Structures in Geometry and Mathematical Physics.

As far as I understand, one of the main philosophy behind the introduction of derived smooth manifolds is to come up with spaces that have the property of fiber product.

In similar setting, Lie groupoids are introduced as generalizations of manifolds, in an attempt to address the question of singularity coming from “quotient”.

I am thinking if there is a notion of groupoid $[X_1\rightrightarrows X_0]$ where both $X_1$ and $X_0$ are derived smooth manifolds instead of just being smooth manifolds. Would this be of any interest to solve some geometric questions? Is there a notion of "derived stack" in the differential geometry setting?

Answer 1

Would this be of any interest to solve some geometric questions. Is there a notion of "derived stack" in the differential geometry setting.

The notion of a derived stack in the setting of differential geometry is defined similarly to derived stacks in analytic or algebraic geometry. Starting from the site given by the opposite category of differential graded ${\rm C}^∞$-rings (appropriately restricted), introduce a Grothendieck topology by adapting the usual definition with open covers, then take simplicial presheaves and perform the left Bousfield localization with respect to Čech covers. Alternatively, simply take the stalkwise weak equivalences, since these coincide with Čech-local equivalence in this case.

For details specific to the case of differential geometry, you can consult the recent paper of Taroyan: Equivalent models of derived stacks.